The base of an equilateral triangle is along the line given by $3\mathrm{x}+4\mathrm{y}=9$ . If a vertex of the triangle is $\left(1,2\right)$  then the length of its side is

The perpendicular distance from vertex  $\left(1,2\right)$ to the base $3x+4y=9$ is height h of the equilateral triangle 
$\begin{array}{c}\mathrm{h}=\left|\frac{3\left(1\right)+4\left(1\right)-9}{\sqrt{3^2+4^2}}\right|\\ =\frac{2}{5}\end{array}$ 
If a  is the length of the side and h is height of an equilateral triangle then  $\mathrm{a}=\frac{2\mathrm{h}}{\sqrt{3}}$
Therefore, the length of the side is $\mathrm{a}=\frac{4}{5\sqrt{3}}=\boxed{0.4618}$